SOFTCOMPUTINGTECHNIQUES











11

CHAPTER-3 SOFT COMPUTING TECHNIQUES Soft Computing is the fusion of methodologies that were designed to model and enable solutions to real world problems, which are not modeled or too difficult to model, mathematically. Soft computing is a consortium of methodologies that works synergistically and provides, in one form or another, flexible information processing capability for handling real-life ambiguous situations [45]. Its aim is to exploit the tolerance for imprecision, uncertainty, approximate reasoning and partial truth in order to achieve tractability, robustness and low-cost solutions [35]. The guiding principle is to devise methods of computation that lead to an acceptable solution at low cost, by seeking for an approximate solution to an imprecisely or precisely formulated problem [46]. Soft computing differs from conventional (hard) computing. Unlike hard computing, it is tolerant of imprecision, uncertainty, partial truth and approximation. In effect, the role model for soft computing is the human mind. Soft Computing is basically optimization technique to find solution of problems which are very hard to answer. 3.1 OPTIMIZATION It is the process of making something better. Optimization is the process of adjusting the inputs to find the minimum or maximum output or result [40]. The Optimization Process is shown in figure 3.1.

Inputor Variables

Function or Process

Output orCost

Figure 3.1 Optimization Process. Optimization is the mechanism by which one finds the maximum or minimum value of a function or process. This mechanism is used in fields such as physics, chemistry, economics, and engineering where the goal is to maximize efficiency,

SOFTCOMPUTINGTECHNIQUES











12

production or some other measure. Optimization can refer to either minimization or maximization; maximization of a function f is equivalent to minimization of the opposite of this function –f. An engineer or scientist conjures up a new idea and optimization improves on that idea. Optimization consists in trying variations on an initial concept and using the information gained to improve on the idea. A computer is the perfect tool for optimization as long as the idea or variable influencing the idea can be input in electronic format. Feed the computer some data and out comes the solution. 3.2 COMBINATORIAL OPTIMIZATION TECHNIQUES Testing problem belongs to the some type of combinatorial optimization problems. The techniques used to tackle combinatorial optimization problems can be classified in two general category, firstly, the exact methods and secondly the approximate (heuristic) methods [41]. Although exact methods granted solution to the problem in hand but not appropriate for real life problems as it requires large computation time seek because of their complex nature, hence the resolution by exact methods is not realistic for large problems, justifying the use of powerful heuristic and meta-heuristics methods. For practical use heuristic methods seek to find high quality solutions (not necessarily optimal) within reasonable computation times [41]. Another type of methods is meta-heuristics which have been applied successfully on large and real life complex problems over the years by different researchers has provided fruitful results [42 - 43]. Classification of common search methodologies are shown in figures 3.2 and 3.3. 3.3 NEURAL NETWORKS (NNS) There are millions of very simple processing elements or neurons in the brain, linked together in a massively parallel manner. This is believed to be responsible for the human intelligence and discriminating power [35]. Neural Networks are developed to try to achieve biological system type performance using a dense interconnection of simple processing elements analogous to biological neurons. Neural Networks are information driven rather than data driven [47]. Typically, there are at least two layers, an input layer and an output layer. One of the most common networks is the Back Propagation Network (BPN) which consists of an input layer, and an output layer with one or more intermediate hidden layers [41].

SOFTCOMPUTINGTECHNIQUES











13

Neural Networks are trained to perform a particular function by adjusting the values of the connections (weights) between elements using a set of examples before they can be employed to the actual problem. Commonly neural networks are adjusted, or trained, so that a particular input leads to a specific target output [44] which is shown in figure 3.4. The method used to generate the examples to train the network and the training algorithm employed has a significant impact on the performance of the neural network-based model. One of the training algorithms used is the BackPropagation (BP) algorithm. This algorithm aims to reduce the deviation between the desired objective function value and the actual objective function value [41].

Optimization

Continuous

Combinatorial

Linear Quadratic

LocalMethod

NonͲLinear

Approximate Method

GlobalMethod

Heuristic

ChemicalMethod Meta Heuristic

PopulationBased NeighborhoodBased

Figure 3.2 Classifications of Common Search Methodologies

ExactMethod

SOFTCOMPUTINGTECHNIQUES











14

Metaheuristics

Neighborhood BasedAlgorithm

Simulated Annealing

Population BasedAlgorithm

Swarm Intelligence

TabuSearch

AntColony Optimization

ParticleSwarm Optimization

Evolutionary computation

Evolutionary programming

Genetic

Algorithm

Genetic Programming

Evolutionary Strategies

Differential

Algorithm

Figure 3.3 Classifications of Common Meta-heuristics.

The performance of this very slow and mostly trapped in local optima, there are another training algorithms which has faster coverage speed and tries to avoid to struck in local optima like Delta-bar-delta (DBD).

SOFTCOMPUTINGTECHNIQUES











15

Target

Input

NNincluding connections(Weights) betweenneurons



Output

Compare

AdjustWeights

Figure 3.4 Neural Networks Limitations The major issues of concern today are the scalability problem, testing, verification and integration of neural network systems into the modern environment. Neural network programs sometimes become unstable when applied to larger problems. The defense, nuclear and space industries are concerned about the issue of testing and verification. The mathematical theories used to guarantee the performance of an applied neural network needs development. The solution for the time being may be to train and test these intelligent systems much as we do for humans. Also there are some more practical problems like: the operational problem encountered when attempting to simulate the parallelism of neural networks. Since the majority of neural networks are simulated on sequential machines, giving rise to a very rapid increase in processing time requirements as size of the problem expands. Networks function as "Black Boxes" whose rules of operation are completely unknown. 3.4 GENETIC ALGORITHMS (GAs) The basic concepts were developed by Holland [8], while the practicality of using the GA to solve complex problems was demonstrated in [49, 51]. Genetic Algorithms (GAs) is a soft computing approach. GAs are general-purpose search algorithms, which use principles inspired by natural genetics to evolve solutions to problems [48]. As one can guess, genetic algorithms are inspired by Darwin's theory about evolution. They have been successfully applied to a large number of scientific and engineering problems, such as optimization, machine learning, automatic programming, transportation problems, adaptive control etc.

SOFTCOMPUTINGTECHNIQUES











16

GA starts off with population of randomly generated chromosomes, each representing a candidate solution to the concrete problem being solved and advances towards better chromosomes by applying genetic operators based on the genetic processes occurring in nature. So far, GAs had a great measure of success in search and optimization problems due to their robust ability to exploit the information accumulated about an initially unknown search space. Particularly GAs specialize in large, complex and poorly understood search spaces where classic tools are inappropriate, inefficient or time consuming. As mentioned, the GA's basic idea is to maintain a population of chromosomes. This population evolves over time through a successive iteration process of competition and controlled variation. Each state of population is called generation. Associated with each chromosome at every generation is a fitness value, which indicates the quality of the solution, represented by the chromosome values. Based upon these fitness values, the selection of the chromosomes, which form the new generation, takes place. Like in nature, the new chromosomes are created using genetic operators such as crossover and mutation. Mechanism of GA The fundamental mechanism consists of the following stages which is also described in figure 3.5. 1. Generate randomly the initial population. 2. Select the chromosomes with the best fitness values. 3. Recombine selected chromosomes using crossover and mutation operators. 4. Insert offspring into the population. 5. If a stop criterion is satisfied, return the chromosome(s) with the best fitness. Otherwise, go to Step 2. In GA, the population is defined to be the collection of individuals. A population is a generation that undergoes under changes to produce new generation. Like nature, GAs have also collection of several members to make population healthy. A chromosome that is a collection of genes is correspondence to individual of population. Each individual chromosome represents a possible solution to the optimization problem. The dimension of the GA refers to the dimension of the search space which equals the number of genes in each chromosome.

SOFTCOMPUTINGTECHNIQUES







Selection

Initialize Population



17

SelectIndividualsforMating

Crossover

MateIndividuals

MutateOffspring

Mutation

InsertInto Population

No



InsertOffspringintoPopulation

Yes Criteria

Answer

Satisfied?

Figure 3.5 Fundamental Mechanism of Simple Genetic Algorithm i)

Representation of Chromosomes The representation of chromosomes in GAs has very deep impact on the

performance of GA-based function. There are different methods of representation of chromosomes like binary encoding, value encoding, permutation encoding, tree encoding etc. The most commonly used encoding is binary encoding proposed by Holland [8]. In this method, the value of individual is encoded as bit string consists of binary values either 0 or 1. Each chromosome of population consists of same length of binary string [36]. Suppose a program has two inputs X and Y having value 8 and 6 respectively and length of binary string is 5. Then, X and Y can be represented as shown in table 3.1.

SOFTCOMPUTINGTECHNIQUES











18

Table 3.1 Binary Encoding Chromosome

Value

Binary Encoding

X

8

01000

Y

6

00110

ii)

Fitness The fitness function is the bone marrow of every optimization problem. If

fitness function is not good, then no optimization methods help in finding the solution. Therefore selection of fitness function is crucial issue. The fitness function provides a way for the GA to analyze the performance of each chromosome in the population. The GA assesses solutions for their quality according to the information produced by this unit and not by using direct information about their structure. Given a particular chromosome, the fitness function returns a single value, which represents the merit of the corresponding solution to the problem. Fitness evaluation functions might be complex or simple depending on the optimization problem at hand. iii)

Selection The selection operator selects chromosomes from the current generation to be

parents for the next generation. The problem is how to select the chromosomes/ individuals. According to Darwin’s theory of evolution the best ones survive to create new offspring. Parents are selected in pairs. There are various types of selection methods which are use to select the chromosomes: •

Uniform Selection In this selection method, parents are selected at random from a uniform

distribution using the expectations and number of parents. This results in an undirected search. Uniform selection is not a useful search strategy, but it can be used to test the genetic algorithm. x

Roulette Wheel Selection This is a stochastic method, where individual having higher fitness value have

more chances to select. This is the most common selection method used in GAs. Roulette wheel selection is shown in figure 3.6.

SOFTCOMPUTINGTECHNIQUES











19

Chromosome1 Chromosome2 Chromosome3 Chromosome4 Chromosome5

Figure 3.6 Roulette Wheel Selections The algorithms of roulette wheel selection is given as a. Do summation of fitness of all chromosomes and store this vale into S b. Generate random number R between 0 and S. c. Go through the population and sum the fitness from 0 - S. When the S is greater than R, stop and return the chromosome where you are. d. Repeat the process from b to c until you get number of chromosomes equal to the size of population [36]. •

Tournament Selection Tournament selection provides selection pressure by holding a tournament

among s competitors, where s is tournament size. The winner of the tournament is the individual with highest fitness of the s tournament competitors. Then winner is inserted into mating pool. The mating pool comprise of tournament winner, has a higher fitness value than average population fitness [50]. The function selects each parent by choosing individuals at random, the number of which you can specify by tournament size, and then choosing the best individual out of that set to be a parent. iv)

Crossover After reproduction simple crossover may proceed in two steps. First, members

of the newly reproduced strings in the mating pool are mated at random. Second each

SOFTCOMPUTINGTECHNIQUES











20

pair of chromosomes undergoes crossing over. Basically there are different types of crossing over which is explained as: Single Point Crossover Crossover position is randomly selected between one and (L-1), where L is the length of chromosome and two parents are crossed at that point. In this crossover, first child is identical to first parent up to the crossing point and identical to the second parent after the crossover point [62]. Figure 3.7 shows single point crossover.

=

Figure 3.7 Single Point Crossover Two Point Crossover In two points crossover, two crossover positions are randomly selected between one and (L-1), where L is the length of chromosome and two parents are crossed at those points. Two point crossover is shown in figure 3.8.

Figure 3.8 Two Point Crossovers

SOFTCOMPUTINGTECHNIQUES

x











21

Uniform Crossover

Uniform crossover is shown in figure 3.9. In uniform crossover, bits are randomly copied from the first or second parents [36]. 1100 1011 0011 +

1110 1000 0000 =

0110 1000 1100

0100 1011 1111

Figure 3.9 Uniform Crossover The amount of crossover is controlled by crossover probability Pc which is defined as the ratio of the number of offspring produced in each generation to the population size. A higher crossover probability allows exploration of more of the solution space and reduces the chances of settling for a false optimum. A lower crossover probability enables exploitation of existing individuals in the population that have relatively high fitness [62]. v) Mutation Mutations are global searches. A probability of mutation is again predetermined before the algorithm is started which is applied to each individual bit of each offspring chromosome to determine if it is to be inverted. As new individuals are generated each character mutated with given probability Pm. In simple GA, mutation means a 1 to a 0 and vice versa with mutation probability [48]. Figure 3.10 shows stages of chromosome before and after mutation. Before Mutation

010100010010

After Mutation

011100000010

Figure 3.10 Mutation Operator The mutation probability Pm controls the rate at which new gene values are introduced into the population. If it is too small, many gene values that would have been useful are never tried out. If it is too high, too much random perturbation will occur and the offspring will lose their resemblance to the parents [62]. The mutation operator plays a secondary role in simple GA. Mutation rates are smaller in natural populations, leading us to conclude that mutation is appropriately considered as a secondary mechanism of genetic algorithm adaptation.

SOFTCOMPUTINGTECHNIQUES











22

3.5 PARTICLE SWARM OPTIMIZATION (PSO) Although GAs provides good solution but they not keep information about the best solution in the whole community. This strategy extends search by the introduction of memory. In this optimization, along with the local best solution, a global best solution is also stored somewhere in the memory, so that all particles not trapped into local optima but moves to global optima. PSO is an algorithm developed by Kennedy and Eberhart [16] that simulates the social behaviors of bird flocking or fish schooling and the methods by which they find roosting places, foods sources or other suitable habitat. The algorithm maintains a population potential where each particle represents a potential solution to an optimization problem. The PSO algorithm works by simultaneously maintaining several candidate solutions in the search space. During each iteration of the algorithm, each candidate solution is evaluated by the objective function being optimized, determining the fitness of that solution. Each candidate solution can be thought of as a particle “flying” through the fitness landscape finding the maximum or minimum of the objective function. Initially, the PSO algorithm chooses candidate solutions randomly within the search space. The initial state of a four-particle PSO algorithm seeking the global maximum in a one-dimensional search space is shown in figure 3.11. The search space is composed of all the possible solutions along the x-axis. The curve denotes the objective function. PSO algorithm has no knowledge of the underlying objective function. Thus has no way of knowing if any of the candidate solutions are near to or far away from a local or global maximum. PSO algorithm uses the objective function to evaluate its candidate solutions and operates upon the resultant fitness values. Each particle maintains its position, composed of the candidate solution and its evaluated fitness and its velocity. Additionally, it remembers the best fitness value it has achieved thus far during the operation of the algorithm, referred to as the individual best fitness and the candidate solution that achieved this fitness, referred to as the individual best position or individual best candidate solution. Finally, PSO algorithm maintains the best fitness value achieved among all particles in the swarm. It is called the global best fitness. The candidate solution that achieved this fitness is called the global best position or global best candidate solution.

SOFTCOMPUTINGTECHNIQUES











23

Figure 3.11 Initial PSO State The PSO algorithm consists of just few steps, which are repeated until some stopping condition is met. The steps are as follow: 1. Initialize the population, location and velocity. 2. Evaluate the fitness of the individual particle (Pbest). 3. Keep track of the individual highest fitness (Gbest). 4. Modify velocity based on Pbest and Gbest location. 5. Update the particle position. 6. Terminate if condition is met else go to step 3. The first two steps are fairly trivial. Fitness evaluation is conducted by supplying the candidate solution to the objective function. Individual, global best fitnesses and positions are updated by comparing the newly evaluated fitnesses against the previous individual and global best fitnesses, and replacing the best fitnesses and positions as necessary. The velocity and position update step is responsible for the optimization ability of the PSO algorithm. The flowchart of PSO algorithm is shown in figure 3.12.

SOFTCOMPUTINGTECHNIQUES











24

Begin

Initializetheparticlespositionand velocity

Calculatethefitnessofeveryparticle

N

Updatetheparticlespositionand velocity Criteriasatisfied

Y Generatereportandthenstop

Figure 3.12 Flow Chart of PSO In the basic PSO technique, suppose that the search space is d-dimensional [21] 1. Each member is called particle, and each particle (i-th particle) is represented by d dimensional vector and described as x i = [xi1, x i2,...,x id]. 2. The set of n particle in the swarm are called population and described as pop=[x1, x2,...,xd]. 3. The best previous position for each particle (the position giving the best fitness value) is called particle best and described as pbi =[pbi1, pbi2,...,pbid]. 4. The best position among all of the particle best position achieved so far is called global best and described as gbi =[gbi1, gbi2,...,gbid]. 5. The rate of position change for each particle is called the particle velocity and described as Vi =[Vi1, Vi2,...,Vid]. At iteration k the velocity for d-dimension of i-particle is updated by:

Vidk 1

wVidk  c1r1 ( pbidk  xidk )  c2 r2 ( gbidk  xidk )

(1)

SOFTCOMPUTINGTECHNIQUES











25

Where i= 1,2,..,n and n is the size of population, w is the inertia weight, c1 and c2 are the acceleration constants, and r1 and r2 are two random values in range [0,1]. 6. The i-particle position is updated by:

x idk  1

x idk  V idk  1

(2)

Each of the three terms of the velocity update equation 1 has different roles in the PSO algorithm. The first term wVKid is the inertia component, responsible for keeping the particle moving in the same direction it was originally heading. The value of the inertial coefficient w is typically between 0.8 and 1.2 which can either dampen the particle’s inertia or accelerate the particle in its original direction. Generally, lower values of the inertial coefficient speed up the convergence of the swarm to optima, and higher values of the inertial coefficient encourage exploration of the entire search space. The second term c1r1(pbkid - xkid) called the cognitive component, acts as the particle’s memory, causing it to tend to return to the regions of the search space in which it has experienced high individual fitness. The cognitive coefficient c1 is usually close to 2 and affects the size of the step the particle takes toward its individual best candidate solution. The third term c2r2(gbkid - xkid) called the social component, causes the particle to move to the best region the swarm has found so far. The social coefficient c2 is typically close to 2 and represents the size of the step the particle takes toward the global best candidate solution gbkid the swarm has found up until that point. The random values r1 in the cognitive component and r2 in the social component cause these components to have a stochastic influence on the velocity update. This stochastic nature causes each particle to move in a semi-random manner heavily influenced in the directions of the individual best solution of the particle and global best solution of the swarm. PSO Algorithm to automatically generate test cases for the given program defines: Step 1: (Initialization): Set the iteration number k=0. Generate randomly n particles, xi , i = 1, 2,..., n, where x i = [xi1, xi2,...,x id]. and their initial velocities Vi =[Vi1, Vi2,...,Vid]. Evaluate the

SOFTCOMPUTINGTECHNIQUES











26

evaluation function for each particle eval (x i) using fitness function. If the constraints are satisfied, then set the particle best PB i = x i and set the particle best which give the best objective function among all the particle bests to global best gb. Else, repeat the initialization. Step 2: Update iteration counter k=k+1. Step 3: Update velocity using Eq. (1). Step 4: Update position using Eq. (2). Step 5: Update particle best: If eval i (xki ) > eval i (pbkí1 i ) then pbki = xki Else pbki = pbk -1i Step 6: Update global best: eval(gbk) = max(eval i (pbkí1 i )) If eval(gbk) > eval(gbkí1) then gbk = gbk Else gbk = gbk -1 Step 7: (Stopping criterion): If the number of iteration exceeds the maximum number iteration or accumulated coverage is 100% then stop, otherwise go to step 2.. 3.6 ANT COLONY OPTIMIZATION (ACO) The idea of ant colony optimization is as its name suggests, inspired from the ant colonies. Ant Colony Optimization (ACO) is a population-based, general search technique for the solution of difficult combinatorial problems, which is inspired by the pheromone trail laying behavior of real ant colonies [73]. Each ant moves along some unknown path in search of food and while it goes it leaves behind a trail of what is known as pheromone. The special feature of this pheromone is that it evaporates with time such that as time proceeds, the concentration of the pheromone decreases on any given path. Now it’s obvious that the path with maximum pheromone is the one that has been traversed the most recently or in fact by most number of ants and hence the most desirable for following ant [74]. The first ACO technique is known as Ant System [75] and it was applied to the traveling salesman problem. This work was further carried by Dorigo, Di Caro, Blum etc [75-79]. Initial attempts at an ACO

SOFTCOMPUTINGTECHNIQUES











27

algorithm were not very satisfying until the ACO algorithm was coupled with a local optimizer. One problem is premature convergence to a less than optimal solution this is because too much virtual pheromone was laid quickly. To avoid this stagnation, pheromone evaporation is implemented. In other words, the pheromone associated with a solution disappears after a period of time [40]. In ACO, a set of software agents called artificial ants search for good solutions to a given optimization problem. The choice of a heuristic technique is quite justified, as the use of any classic greedy approach shows very poor results [80]. The use of ant colony optimization is best for the graph based problems [74]. To apply ACO, the optimization problem is transformed into the problem of finding the best path on a weighted graph. The artificial ants (hereafter ants) incrementally build solutions by moving on the graph. The solution construction process is stochastic and is biased by a pheromone model, that is, a set of parameters associated with graph components (either nodes or edges) whose values are modified at runtime by the ants [73]. Algorithm for ACO is shown in figure 3.13. Step 1: Initialization – Initialize the pheromone trail Step 2: Iteration – For each Ant Repeat – Solution construction using the current pheromone trail – Evaluate the solution constructed – Update the pheromone trail – Until stopping criteria Figure 3.13 A Generic Ant Colony Algorithm The ACO is a natural for the traveling salesperson problem. It begins with a number of ants that follow a path around the different cities. Each ant deposits a pheromone along the path. The algorithm begins by assigning each ant to a randomly selected city. The next city is selected by a weighted probability that is a function of the strength of the pheromone laid on the path and the distance of the city. The probability that ant k will travel from city m to city n is given by ௞ ܲ௠௡ ൌ

௔ ௕ ߬௠௡ Ȁ݀௠௡  ௤ ௕ σ௤ ߬௠௡ Ȁ݀௠௡

SOFTCOMPUTINGTECHNIQUES











Where IJ = pheromone strength q = cities on tour k that come after city m a = pheromone weighting; when a = 0, closest city is selected b = distance weighting; when b = 0, distance between cities is ignored Short paths with high pheromone have the highest probability of selection [40].

28